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Metamath Proof Explorer

Theorem llnneat

Description: A lattice line is not an atom. (Contributed by NM, 19-Jun-2012)

Ref Expression
Hypotheses llnneat.a 
|- A = ( Atoms ` K )
llnneat.n 
|- N = ( LLines ` K )
Assertion llnneat 
|- ( ( K e. HL /\ X e. N ) -> -. X e. A )

Proof

Step Hyp Ref Expression
1 llnneat.a 
 |-  A = ( Atoms ` K )
2 llnneat.n 
 |-  N = ( LLines ` K )
3 hllat 
 |-  ( K e. HL -> K e. Lat )
4 eqid 
 |-  ( Base ` K ) = ( Base ` K )
5 4 2 llnbase 
 |-  ( X e. N -> X e. ( Base ` K ) )
6 eqid 
 |-  ( le ` K ) = ( le ` K )
7 4 6 latref 
 |-  ( ( K e. Lat /\ X e. ( Base ` K ) ) -> X ( le ` K ) X )
8 3 5 7 syl2an 
 |-  ( ( K e. HL /\ X e. N ) -> X ( le ` K ) X )
9 6 1 2 llnnleat 
 |-  ( ( K e. HL /\ X e. N /\ X e. A ) -> -. X ( le ` K ) X )
10 9 3expia 
 |-  ( ( K e. HL /\ X e. N ) -> ( X e. A -> -. X ( le ` K ) X ) )
11 8 10 mt2d 
 |-  ( ( K e. HL /\ X e. N ) -> -. X e. A )